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i'm an undergrad math student and have lots of problems with group theory.i hope you'll all help me enjoy group theory...

my teacher put forward these question last week and i've been breaking my head over them without much success :

1. let G be a finite cyclic group of order p^n, p being prime and n >=0. if H and K are subgroups of G then show that either H contains K or K contains H.

i started out supposing the contrary but i wonder if i'm on the right track. i don't think it helps.

2.if G is a group of order 30 show that G has atmost 7 distinct subgroups of order 5.

can i say this : let H be a subgroup of order 5 then the number of distinct left cosets of H in G is 6. so are we done?!

3.let G be a group such that intesection of all subgroups of G different from {e}. then prove that every element of G has finite order.

4. give an example to show that a subgroup of index 3 may not be a normal subgroup of G.

thanks again for the help.

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# Subgroup math help

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